Equivalence between the pair correlation functions of primes and of spins in a two-dimensional Ising model with randomly distributed competing interactions

TL;DR

This paper establishes a surprising equivalence between the pair correlation functions of primes, spins in a 2D Ising model with random interactions, and zeros of the Dirichlet function, revealing deep connections across number theory and statistical physics.

Contribution

It proves the equivalence of correlation functions across primes, spins, and zeros of the Dirichlet function, linking number theory and physics in a novel way.

Findings

Pair correlation function of spins is positive at all temperatures.

Correlation functions of spins are equivalent to those of energy levels.

Prime pair correlations for even q are larger than zero.

Abstract

In this work, we prove the equivalence between the pair correlation functions of primes, and of spins in a two-dimensional (2D) Ising model with a mixture of ferromagnetic and randomly distributed competing interactions. At first, we prove that the correlation function between a pair of spins in a distance l within the 2D Ising model is larger than zero at whole temperature region. Second, we prove that the pair correlation function of spins in the model is equivalent to the pair correlation function of its energy levels. Third, we prove that the energy-energy correlation function of the model is equivalent to the pair correlation function of nontrivial zeros of the Dirichlet function (including the Riemann zeta function). Fourth, we prove that the pair correlation function between the nontrivial zeros of the Dirichlet function is equivalent to the correlation function between a pair of primes p and p+q for every even q. In a conclusion, we have proven that the pair correlation function of primes p and p+q for every even q is larger than zero.